Mechanics 2: Linear and Rotational Dynamics - 3D Math Primer for Graphics and Game Development - page 611

Back to our thought experiment with the mass attached to the rotating
disk. What if we fix the lever arm l, and instead vary r, the radial distance
between the mass and the pivot? The same principle of the lever is at work,
but in reverse. The force (and therefore the tangential acceleration a ⊥ )
experienced by the mass will be inversely proportional to r. Said another
way, the effective inertia of the mass—its resistance to linear acceleration—
is proportional to r. But what about the ability of the apparatus to resist
angular acceleration? How does its moment of inertia change as we vary r?
The moment of inertia is not proportional to r, it is proportional to r 2 ! To
see why, consider that if we increase l and r by the same factor, then the
tangential acceleration a ⊥ experienced by the mass is unchanged. However,
at this increased radius, the same tangential acceleration now corresponds
to a reduced angular acceleration, due to the relation α = a ⊥ /r.
In summary, the moment of inertia of an object, which must be mea-
sured relative to some particular pivot (in this case, it's the fixed pivot
point, but for a rigid body we usually measure it relative to its center
of mass), quantifies the degree to which the object will resist angular ac-
celeration about that pivot. The moment of inertia J of a point mass is
proportional to its mass and proportional to the square of the distance from
the mass to the pivot.

Moment of Inertia of a Point Mass in the Plane

J = mr 2 .

Now imagine that the disk in our thought experiment has multiple
masses placed on it. Each of these masses contributes to the disk's re-
sistance to rotation, and their contribution is the same, regardless of where
any force is applied. To compute the moment of inertia of an arbitrary
rigid body, we break up the object into “mass elements” such that for each
element we know the mass m i and radial distance to the center of mass r i .
We then sum up the moments of inertia of each individual mass element:

Moment of inertia of a
rigid body

m i r i .

J = J i = (12.25)

i i

Let's work through a few instructive examples.